Journal of Biomechanics 47 (2014) 1853–1860
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Quantitative analysis of benign paroxysmal positional vertigo fatigue under canalithiasis conditions F. Boselli a,n, L. Kleiser a, C.J. Bockisch b,c,d, S.C.A. Hegemann b, D. Obrist a a
Institute of Fluid Dynamics, ETH Zurich, 8092 Zurich, Switzerland Department of Otorhinolaryngology, Head and Neck Surgery, University Hospital Zurich, 8092 Zurich, Switzerland c Department of Neurology, University Hospital Zurich, 8092 Zurich, Switzerland d Department of Ophthalmology, University Hospital Zurich, 8092 Zurich, Switzerland b
art ic l e i nf o
a b s t r a c t
Article history: Accepted 11 March 2014
In our daily life, small flows in the semicircular canals (SCCs) of the inner ear displace a sensory structure called the cupula which mediates the transduction of head angular velocities to afferent signals. We consider a dysfunction of the SCCs known as canalithiasis. Under this condition, small debris particles disturb the flow in the SCCs and can cause benign paroxysmal positional vertigo (BPPV), arguably the most common form of vertigo in humans. The diagnosis of BPPV is mainly based on the analysis of typical eye movements (positional nystagmus) following provocative head maneuvers that are known to lead to vertigo in BPPV patients. These eye movements are triggered by the vestibulo-ocular reflex, and their velocity provides an indirect measurement of the cupula displacement. An attenuation of the vertigo and the nystagmus is often observed when the provocative maneuver is repeated. This attenuation is known as BPPV fatigue. It was not quantitatively described so far, and the mechanisms causing it remain unknown. We quantify fatigue by eye velocity measurements and propose a fluid dynamic interpretation of our results based on a computational model for the fluid–particle dynamics of a SCC with canalithiasis. Our model suggests that the particles may not go back to their initial position after a first head maneuver such that a second head maneuver leads to different particle trajectories causing smaller cupula displacements. & 2014 Elsevier Ltd. All rights reserved.
Keywords: BPPV Canalithiasis Semicircular canals (SCCs) Endolymph flow Nystagmus Method of Fundamental Solutions Force Coupling Method
1. Introduction The semicircular canals (SCCs) of the inner ear are the primary human sensors for head rotation. The SCCs are three, approximately mutually orthogonal, slender ducts which span an angle of approximately 2501 and merge into a bigger chamber called utricle (Fig. 1). One end of each SCC has a larger cross-section, the ampulla, that is plugged by a gelatinous flexible structure, the cupula. The fluid which fills the SCC, the endolymph, lags the moving canal walls during head rotation such that it deflects the cupula and the embedded hair cell bundles. This displacement and the resulting afferent signals are proportional to the head velocity, and trigger compensatory eye movements [the Vestibulo-Ocular Reflex (VOR)], which rotate the eyes opposite to the head rotation, stabilizing the eye in space and reducing visual blur. If the eye moves too far from a central position, a fast corrective movement (saccade) returns the eyes to their central position. During n
Corresponding author. E-mail addresses: [email protected] (F. Boselli), [email protected] (L. Kleiser), [email protected] (C.J. Bockisch), [email protected] (S.C.A. Hegemann), [email protected] (D. Obrist). http://dx.doi.org/10.1016/j.jbiomech.2014.03.019 0021-9290/& 2014 Elsevier Ltd. All rights reserved.
prolonged stimulation, the alternating slow compensatory and fast resetting movements are called nystagmus. A mechanical dysfunction of the SCCs, mainly of the posterior canal (PC), can lead to Benign Paroxysmal Positional Vertigo (BPPV), arguably the most common form of vertigo in humans (Baloh et al., 1989). This dysfunction is often associated with canalithiasis, a condition where free-floating particles (canaliths) reside in the SCC. Provocative head maneuvers (HM), e.g. when tilting the head backward to take a book from the top shelf, lift the particles to a higher position from which they can then settle under the action of gravity. This can lead to a flow similar to that induced by head rotations (Hall et al., 1979; Epley, 2001). The resulting post-rotatory cupula displacement triggers the misleading spinning sensation which is experienced during BPPV. The vertigo following the HM is revealed by a ‘positional’ nystagmus. This understanding of BPPV is supported by the intra-operative observation of particles in the membranous PC (Parnes and McClure, 1992), by several in-vivo models based on ampullaryafferents measurements (Suzuki et al., 1996; Inagaki et al., 2006; Rajguru and Rabbitt, 2007; Valli et al., 2009), and by several theoretical (House and Honrubia, 2003; Squires et al., 2004; Rajguru et al., 2004; Obrist and Hegemann, 2008; Boselli, 2012)
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Fig. 1. The membranous labyrinth (blue) and the bony labyrinth (yellow) of the inner ear (adapted from Obrist et al., 2010). The width of the membranous anterior (AC), posterior (PC), and horizontal (HC) canals is exaggerated for better visibility. In reality their width is only about 5% of the diameter of the bony canals (Curthoys et al., 1977). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
and in vitro (Obrist et al., 2010) models. In the case where the particles stick to the cupula rather than float in the SCC, it is common to talk about “cupulolithiasis” rather than canalithiasis. Cupulolithiasis and canalithiasis can coexist and lead to a very similar nystagmus (Cohen and Sangi-Haghpeykar, 2010). It is typical for BPPV that the positional nystagmus is generally weaker when the HM is repeated (Dix and Hallpike, 1952). This intricate phenomenon, known as BPPV fatigue, is usually attributed to the disintegration of particle lumps into smaller parts (Parnes and McClure, 1992; Parnes and Price-Jones, 1993; Parnes et al., 2003). However, previous theoretical studies do not support this mechanical interpretation of BPPV fatigue. Squires et al. (2004) concluded that a train of particles settling in a SCC would lead to a higher cupula displacement than a single particle of equal mass. Similarly, Rajguru and Rabbitt (2007) and Obrist and Hegemann (2008) repeated simulations of a HM by increasing the number of particles while keeping the total mass constant. This resulted in an increased cupula displacement rather than in ‘fatigue’. These theoretical models, however, are based on the Stokes drag force of isolated spheres and cannot account for the interaction between particles. In contrast, Boselli (2012) introduced a model based on the force coupling method (Lomholt et al., 2002) which can account for the finite size of the particles and their hydrodynamic interaction. This model was able to predict up to a 40% reduction (i.e. a fatigue) of the cupula displacement but only if the particle lump before the first HM blocks almost the whole canal lumen. For this particle size, however, the particle-driven flow is dominant already during the HM, which is in contrast to typical nystagmus measurements where the particledriven flow dominates only after the HM. Furthermore, these numerical experiments assume the resting particle position before the first and the second HM to be about the same. In the present work, we quantify the fatigue of the nystagmus in several patients and apply the model of Boselli (2012) to provide an interpretation of our nystagmus measurements based on the redistribution of particles in the SCC due to a HM.
2. Materials and methods 2.1. Nystagmus measurements The fatigue of the positional BPPV nystagmus is discussed only qualitatively in the literature. Therefore, we started by analyzing clinical nystagmus measurements of nine patients and discuss how the nystagmus changes for repeated HMs. The experimental protocol was approved by the Ethics Committee of the Canton of Zurich, Switzerland, and all patients provided written consent after the experimental procedure was explained.
Patients sat in a motorized chair that rotated the subject about an axis that intersected the head orthogonally to the plane of the posterior canal. In a first head maneuver (HM1), the rotation was backwards by 1201. The rotation was done with a constant acceleration of 101/s2 for half of the movement (601) and then decelerated with 101/s2 for the second half of the movement. This yields a triangular velocity profile with a peak of about 351/s and a total duration of about 7 s (Fig. 2). The HM1 was then reversed, and the patient brought back upright. A second head maneuver (HM2), identical to HM1, was then applied in order to assess the fatigue of the nystagmus. Before applying any rotation, we waited until the patient's nystagmus and feelings of dizziness stopped (typically at least 60 s). The orientation of the SCC before and after these rotations is illustrated in Fig. 4. For two patients, a third HM was also applied after reversing HM2. Three-dimensional eye movements were measured by the scleral search coil technique (Collewijn et al., 1985), and the rotational velocity in the plane of the affected canal was calculated. The nystagmus was measured only during and after the backward rotations. A reference light at gaze straight ahead was turned on at the start of the head rotation, and was turned off at the end. While this light suppressed eye movements during the head movement, distinguishing BPPV related nystagmus from the actual VOR would be difficult in any case. Saccades were removed with an interactive computer program that automatically detected saccades when velocity exceeded a threshold (typically about 201/s, depending upon the noise) above the median eye velocity calculated over a one second window. The automatically marked saccades could be manually adjusted and blink artifacts removed. The median velocity of each slow phase was calculated (Bockisch et al., 2013, 2012), and the resulting velocity trace was smoothed with a locally weighted second degree polynomial (smooth.m with a 10 s span, Mathworks). All patients were diagnosed with posterior canal BPPV on examination and a positive response to a Dix–Hallpike maneuver. The patients had no other neurologic or eye movement disorders. Five patients had BPPV of the left posterior canal, and four of the right posterior canal. 2.2. Numerical model The endolymph flow is modeled by the quasi-steady Stokes equations ∇pðx; tÞ þ μ∇2 uðx; tÞ ¼ f c þ f α þ f p
ð1aÞ
∇ uðx; tÞ ¼ 0
ð1bÞ
with no-slip boundary conditions at the wall; t is the time, x ¼ ðx1 ; x2 ; x3 Þ are the Cartesian coordinates of the rotating reference frame, p and u ¼ ðu1 ; u2 ; u3 Þ are the pressure and the velocity of the fluid, respectively. The inertial force f α ðx; tÞ ρα€ x arises from the angular acceleration α€ of the SCC during the head rotation (e.g. Boselli et al., 2013b; Oman et al., 1987); the force f c is the restoring force of the cupula on the fluid; and f p represents the forces exchanged between the particle and the fluid phases. Cupula model: The force f c is modeled as in the previous works (e.g. Oman et al., 1987) by introducing a pressure difference ΔP c ðtÞ ¼ KV c ðtÞ across the cupula which is proportional to the volume Vc displaced by the HM and/or the particles, where K is the stiffness of the cupula. The volumetric displacement Vc(t) of the cupula Z t ZZ u nc dA dt ð2Þ V c ðtÞ ¼ Ac
0
is computed numerically by time integration of the flow rate at a cross-section Ac of the SCC, where nc is the normal vector on the surface Ac (Boselli et al., 2013b). Fictitious forces: We adopt the force coupling method (FCM) and model each particle as a finite-size monopole (Maxey and Patel, 2001) such that f p becomes np
f p ðx; tÞ ¼ ∑ Fξ ΔM ðx Y ξ ðtÞÞ ξ¼1
ð3Þ
where Y ξ is the center of the ξ-th particle and np is the number of particles. The monopole strength Fξ is given by the sum of the volumetric forces acting on the particle and includes the gravity force, attraction/repulsive forces between the particles, and wall-particle lubrication forces (Dance and Maxey, 2003). The lubrication force, if included, is precomputed for a single particle to match asymptotic solutions (Boselli, 2012). The sliding velocity of the particle at the wall depends on the minimum distance (lubrication gap) that is imposed between the particle and the wall. The force Fξ is distributed over the particle volume by the force envelope ΔM " # −3=2 −ðx−Y ξ Þ2 ΔM x−Y ξ ¼ 2πσ 2M exp ; ð4Þ 2σ 2M pffiffiffiffi with the length scale sM ¼ ap = π similar to the particle radius ap . The fluid domain is extended over the particle volume and the velocity vξ of each particle is given by the weighted average of u at the particle location Z 3 uðx; tÞΔM ðx Y ξ ðtÞÞ d x: ð5Þ vξ ðtÞ ¼ R3
Eqs. (1) and (5) are solved by coupling the multilayer method of fundamental solutions (Boselli et al., 2012) to the FCM as proposed by Boselli et al. (2013a). We
F. Boselli et al. / Journal of Biomechanics 47 (2014) 1853–1860
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Fig. 2. Clinical data: Total velocity of the slow-phase positional nystagmus during ( 7 so t o 0 s) and after (0 s o t o 45 s) the first ( ), the second ( ) and the third HM ( ; only for (d) and (f)). The angular velocity (in degrees per second) of the head ( ) is shown as control. The reference light is switched off at t 1 s. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.) constrain the particle in the slender part of the posterior canal (PC), where we assume the SCC lumen to have a constant radius ascc ¼ 182 μm. The Reynolds number is small such that the local curvature of the canal can be neglected. This allowed us to map the PC geometry to a torus such that we can employ the computationally efficient domain decomposition approach proposed by Boselli et al. (2013a) for a toroidal domain. The mapping is performed by defining the position of each particle and the direction of the forces with respect to the centerline and its tangent vector, respectively. We consider the PC geometry shown in Fig. 4. Its centerline was obtained by interpolation of a projection of the anatomical model by Wang et al. (2006) in the plane of the PC.
3. Results 3.1. Nystagmus measurements Fig. 2 shows eye velocity traces for 6 of our 9 patients. Time is measured with respect to the end of the head maneuver (in green). Eye velocity typically rapidly increased after the head rotation, usually reaching a peak within 10 s of the end of the movement.
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Even-though a fixation light was present during the head movement, eye velocity begins to build up during the head movement in some
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Fig. 3. Model results: Volumetric cupula displacement Vc for the repeated head maneuver illustrated in Fig. 4; np ¼ 1 and ap ¼ 28:4 μm. Dotted lines indicate the end of each head-maneuver. Arrows indicate the peak value (Vmax) of the postrotatory cupula displacement following each head rotation. The green line shows the control (no particles). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
patients. Note, however, that this velocity could be the normal VOR response to the head movement, and not a reaction to particle movement in the canals. Only the eye velocity after the head movement can be definitively attributed to BPPV. Fatigue was clearly observed for six out of nine patients. The observed percentage reduction of the peak velocity of the positional nystagmus (t 40 s) after the second HM varies from 10 to 70% (e.g. Fig. 2(a)–(c)). For one patient, no nystagmus was observed after the second HM (100% reduction; not shown). Interestingly, for two patients (Fig. 2(e) and (f)), the positional nystagmus after the first HM shows multiple peaks. In some cases, fatigue is not clearly observed, but the peak velocity may slightly increase (Fig. 2(d)–(f)). In one of these patients (Fig. 2(d)), the second HM led to a 10% reduction of the peak velocity of the positional nystagmus, while the third HM led to an increase of about 10% (with respect to HM1). For most of the cases, we do not observe any onset latency of the positional nystagmus. This suggests that the particle-driven flow becomes significant already during the slow HM. 3.2. Numerical simulations 3.2.1. Change in the initial position due to SCC geometry The same HMs imposed during the measurements (Fig. 2, green) were implemented in our computational model. The predicted cupula
Fig. 4. Orientation of the PC (green) before (left; patient upright) and after (right) the first (top) and the second (bottom) head maneuver (HM). The red sphere illustrates the position of the particle at the beginning (left) and at the end (right) of the HM. A, B, and C indicate the initial position of the particle before each head rotation (i.e. HM1, reversed HM1, and HM2). Circular arrows indicate the direction of head rotation. Lines illustrate the particle trajectory with arrows indicating the particle motion. The bony labyrinth (topleft; gray) is shown for illustrative purposes only. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
F. Boselli et al. / Journal of Biomechanics 47 (2014) 1853–1860
displacement Vc computed for one particle of radius ap ¼ 28:4 μm is shown in Fig. 3 together with a control (result for a healthy SCC without particles). Note that the considered HM is very slow such that the SCC operates close to its lower corner frequency (Van Buskirk et al., 1976). Therefore, the cupula displacement for the control does not exactly follow the triangular shape of the imposed angular velocity, but we observe a large overshoot toward the end of the HM. When the first HM (HM1) starts, it is reasonable to assume that the particle is close to the ampulla (Fig. 4, position A), at the lowest point of the PC, when the subject is upright. During HM1, the particle is lifted to a higher position from which it settles toward the lower part of the canal, away from the ampulla. In agreement with the common understanding of canalithiasis (Hall et al., 1979; Epley, 2001), this leads to a temporary, particle-driven flow which results in a misleading, ampullofugal (V c o 0) post-rotatory cupula displacement while in fact the head is at rest (Fig. 3). This post-rotatory displacement arises after an onset-latency which is required to compensate for the cupula overshoot observed at the end of the HM for the control (Obrist and Hegemann, 2008). Reversing the HM1 to bring the patient back upright, the particles settle back, leading to a weaker, ampullopetal (V c 4 0), i.e. inhibiting, post-rotatory cupula displacement (Fig. 3; not measured in vivo). The particle may settle at a position (Fig. 4, position B) which is farther from the ampulla than its initial position A. This shift in the initial particle position is favored by the fact that the canal centerline between points A and
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B is almost straight. Starting from position B rather than A, the particle is lifted to a lower position during HM2. Settling from this lower position (Fig. 4), the particle leads to a weaker post-rotatory cupula displacement with a peak value Vmax which is about 70% smaller than what has been observed after HM1 (cf. Fig. 3). This reduction can explain the fatigue of the positional nystagmus observed in our patients. Simulations were repeated for several values of particle mass, size ap and number np, and with and without lubrication force (Table 1). The reduction of Vmax due to repeated HMs ranges between 25% and 84% for the presented simulations, and decreases by increasing either ap, np, or particle mass (Table 1). Increasing the particle-wall friction by adding the lubrication force tends to lift the particles to a higher position and to enhance Vmax. Fatigue is also enhanced (Table 1) because it increases the difference in the height to which the particles are lifted by HM1 and HM2, respectively. To test the effect of small changes in the head position, some simulations were repeated with the SCC rotated by 101 (i.e. position A lower and point B higher at t¼ 0). The particles reach a lower position after HM1 and lead to a slightly lower value of Vmax, while cupula displacement following HM2 is unchanged. This tends to slightly weaken fatigue. In our simulations with multiple particles, the particles line up near position B when HM2 starts (Fig. 5(a), top). However, the actual position of the particles when HM2 starts will depend on the precise SCC orientation and particle–wall interaction.
Table 1 Percentage reduction of the maximum post-rotatory cupula displacement due to a change in the particles initial position from about A (HM1) to about B (HM2). The particles sizes are chosen such that the total particle mass is constant along the diagonals of this table.
np ¼ 1 np ¼ 2 np ¼ 4 np ¼ 8
ap ¼ 14:2 μm
ap ¼ 17:89 μm
– – – 30%
– – 80% –
(a) Results with lubrication force ap ¼ 22:54 μm ap ¼ 28:4 μm – 84% – –
74.1% 73.22% 67.34% –
ap ¼ 35:78 μm
ap ¼ 45:08 μm
71.6% – – –
25% – – –
(b) Results without lubrication force
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ap ¼ 11:27 μm
ap ¼ 14:2 μm
ap ¼ 17:89 μm
ap ¼ 22:54 μm
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– – – – 75%
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t [s] Fig. 5. (a) Different initial positions of the particles for the simulations with np ¼10 and ap ¼ 0:13ascc . (b) Volumetric cupula displacement for the different initial positions defined in (a): , close to the ampulla (position A); , away from ampulla (position B); y, dispersed between position A and B.
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Therefore, we tested another possible scenario where the particles are dispersed along the horizontal segment AB of the canal (Fig. 5 (a)) before HM2 starts. For such a scenario, we present results for np ¼10 and ap ¼ 0:13ascc (Fig. 5(b)). A first HM is simulated by assuming that the particles are initially close to the ampulla about position A (Fig. 5(a), top). The particle-driven flow becomes dominant during the HM, such that no onset-latency is observed. The volumetric cupula displacement shows two welldistinguishable peaks. These are associated with the formation of two groups of particles that detach from the inner wall at two different times (cf. Fig. 4, top right). If we assume the particles to be dispersed between the positions A and B (Fig. 5(a), center) when HM2 starts, the presence of two separate peaks becomes less pronounced in the resulting cupula displacement. This result is similar to what we observe in some of our nystagmus measurements (e.g. Fig. 2(e)), where the double peak was observed only after HM1. Particles positioned about position B (Fig. 5(a)) before the HM lead to a weaker nystagmus because of the larger average initial distance of the particles from the ampulla. 3.2.2. Particle lump disintegration In order to replicate the disintegration of the particle lump suggested by Parnes and McClure (1992), we present an example (Fig. 6(a)) where we add an attraction force between the particles in the form of a Lennard-Jones potential (e.g. Castellanos, 2005). When the particles are at rest, they lump together because of this attraction force. During the first HM, the lump breaks and the particles disperse while settling in the SCC (Fig. 6(b)). The resulting post-rotatory cupula displacement shows a double-peak cupula displacement that resembles the nystagmus velocity in Fig. 2(e). Before HM2 starts, the particles tend to align along the wall in a single file and come to rest around position B. From this position, the particles lead to a weaker cupula displacement after HM2, as for the simulations without attraction force. The lubrication gap for this example was set to 10 6 ap , and the lubrication force was precomputed for ap ¼ 0:13ascc . In order to estimate the effect of particle disintegration alone (with the particles moving back to position A after HM1), we can compare the cupula displacement following HM1 obtained with and without attraction force (Fig. 6(a)). In this example, the attraction force reduces the predicted Vmax. This supports the idea that disintegration of a particle lump alone is not the only explanation for fatigue. Rather the particles have to move further away from the cupula. 3.2.3. Exceptions not leading to fatigue In order to identify conditions that could explain cases where fatigue is not observed, we considered the SCC after a backward rotation by 1201 (e.g. after HM1) and plot the cupula displacement induced by a particle settling from different elevated positions. Examples are shown in Fig. 7. Based on the initial height of the particle, we can identify three types of particle trajectories: type I, II and III. A particle following the type I trajectory settles first along the inner side of the wall before detaching again from the wall and reattaching at the outer side (Fig. 7, trajectories 1–4). A particle following a type II trajectory detaches significantly from the outer wall, but does not slide on the inner wall (Fig. 7, trajectories 6 and 7). A particle following a type III trajectory detaches only slightly from the outer wall and reattaches again at the outer wall, or simply settles along the outer wall without any detachment from the wall (Fig. 7, trajectory 8). Fig. 7(b and c) shows that Vmax does not decrease monotonically with decreasing initial altitude of the particle. Instead, Vmax can increase when the initial altitude is decreased in the narrow range where we observe the transition between type I and
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Fig. 6. (a) Volumetric cupula displacement during and after the first (black) and the second (red) HM for np ¼8 and ap ¼ 15 μm. Solid lines: Simulation with attraction force; dashed line: Simulation without attraction force. (b) Snapshots of the particles during and after HM1 for the simulation with the attraction force. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
type II trajectories (Fig. 7, trajectories 4–6). This suggests that the post-rotatory cupula displacement after HM2 can increase if the particles follow a type I trajectory after HM1 and a type II trajectory after HM2. This provides a possible explanation for the more rare cases where a slight increase of the eye velocity is observed when repeating the HM (cf. Fig. 2(d) and (f)). Furthermore, the time to peak tp (Fig. 7(c)) tends to decrease with decreasing initial altitude for type I trajectories (Fig. 7, trajectories 1–5), while it increases for type II and III trajectories (Fig. 7, trajectories 6–8).
4. Discussion and conclusions We presented nystagmus measurements for repeated head maneuvers (HMs) which provide a quantitative reference for BPPV fatigue. The reduction in cupula displacement leading to
F. Boselli et al. / Journal of Biomechanics 47 (2014) 1853–1860
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Fig. 7. The PC is held fixed in its position at the end of the head maneuver and a particle is released at t¼ 0 from different initial positions which leads to different trajectories (numbered from 1 to 8). The cupula displacement Vc at t ¼0 is set to zero. (a) Particle trajectories; (b) volumetric cupula displacements Vc; (c) peak volumetric cupula displacement Vmax and time to peak tp. No lubrication force is considered; ap ¼ 36:8 μm.
the measured decrease in eye velocity was predicted by means of a numerical model of particle–fluid dynamics in the SCC. The presented results suggest that the fatiguability of the nystagmus can be explained by the fact that a first HM tends to move the particles farther from the ampulla. This is favored by the PC anatomy which may feature a ‘straight’ part close to the ampulla. We also modeled disintegration of a particle lump by introducing an attraction force between particles. This condition was suggested to represent the actual scenario in vivo (Parnes and McClure, 1992). However, fatigue is enhanced when the attraction force is not considered, suggesting that disintegration of particle lumps is not necessary for fatigue and that fatigue is dominated by changes in the position of the particles following a first HM. Fatigue was not observed for 3 out of 10 patients, for which the positional nystagmus was even increased when repeating the HM.
For these patients, the particles might have gone back to their initial position after HM1 (e.g. due to a particular SCC shape). A second possibility is that the particle initial position for the two consecutive HMs is such that the particle trajectory changes from type I to type II trajectory as discussed in Section 3.2.3 (e.g. from trajectory 4 to 6 in Fig. 7), which would lead to an increase of the cupula displacement following HM2. A further possibility (not modeled) is that these patients were initially affected by cupulolithiasis. This would prevent fatigue because the particles cannot move away from the cupula. In such a scenario, if one of the particles detaches from the cupula following HM1, it will generate a canalithiasis on top of the cupulolithiasis (Cohen and SangiHaghpeykar, 2010). This could increase the nystagmus following HM2, because the same particle leads to a bigger transcupular pressure under canalithiasis rather than cupulolithiasis (Squires et al., 2004; Rajguru et al., 2004).
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Besides the actual fatigue, our computational model could reproduce the multiple peaks of the positional nystagmus observed in two of our patients. This feature is associated with the different instants at which the different particles detach from the wall. In some patients the time to peak tp seems to decrease after the first HM. This feature was not predicted by our model. We associate this discrepancy with the fact that our model assumes a uniform radius of the SCC, while the cross sectional area increases toward the cupula of the actual SCC. Squires et al. (2004) showed that a particle does not affect the cupula significantly while settling in a larger region. We can then imagine that the particles in position A settle first along a larger region during HM1 such that their action on the cupula is delayed. In contrast, the particles are already in the narrow part of the SCC (position B) when HM2 starts, and they immediately become effective in displacing the cupula when they start settling such that tp is shorter after HM2 than after HM1. A natural next step for our model would be the employment of patient-specific geometries, including the non-planarity of the SCC. Recent advances in the field of vestibular anatomy (Bradshaw et al., 2010) suggest that this might become possible in the near future. Conflict of interest statement All authors declare that there are no conflicts of interest. Acknowledgment The work of F.B. was supported by the Swiss National Science Foundation (SNF # 200021-116575). References Baloh, R.W., Sloane, P.D., Honrubia, V., 1989. Quantitative vestibular function testing in elderly patients with dizziness. Ear Nose Throat J. 68, 935–939. Bockisch, C.J., Khojasteh, E., Straumann, D., Hegemann, S.C.A., 2012. Development of eye position dependency of slow phase velocity during caloric stimulation. PLoS ONE 7, e51409, http://dx.doi.org/10.1371/journal.pone.0051409. Bockisch, C.J., Khojasteh, E., Straumann, D., Hegemann, S.C.A., 2013. Eye position dependency of nystagmus during constant vestibular stimulation. Exp. Brain Res. 226, 175–182, http://dx.doi.org/10.1007/s00221-013-3423-6. Boselli, F., 2012. Fluid Dynamics of the Balance Sense: A Computational Study Based on the Multilayer Method of Fundamental Solutions (Ph.D. thesis). ETH Zurich, Zurich, Switzerland. Diss. ETH No. 20576. Boselli, F., Obrist, D., Kleiser, L., 2012. A multilayer method of fundamental solutions for Stokes flow problems. J. Comput. Phys. 231, 6139–6158, http://dx.doi.org/ 10.1016/j.jcp.2012.05.023. Boselli, F., Obrist, D., Kleiser, L., 2013a. A meshless boundary method for Stokes flows with particles: application to canalithiasis. Int. J. Numer. Methods Biomed. Eng. 29, 1176–1191, http://dx.doi.org/10.1002/cnm.2564. Boselli, F., Obrist, D., Kleiser, L., 2013b. Vortical flow in the utricle and the ampulla: a computational study on the fluid dynamics of the vestibular system. Biomech. Model. Mechanobiol. 12, 335–348, http://dx.doi.org/10.1007/s10237-012-0402-y. Bradshaw, A., Curthoys, I., Todd, M., Magnussen, J., Taubman, D., Aw, S., Halmagyi, G., 2010. A mathematical model of human semicircular canal geometry: a new
basis for interpreting vestibular physiology. J. Assoc. Res. Otolaryngol. 11, 145–159, http://dx.doi.org/10.1007/s10162-009-0195-6. Castellanos, A., 2005. The relationship between attractive interparticle forces and bulk behaviour in dry and uncharged fine powders. Adv. Phys. 54, 263–376, http://dx.doi.org/10.1080/17461390500402657. Cohen, H.S., Sangi-Haghpeykar, H., 2010. Nystagmus parameters and subtypes of benign paroxysmal positional vertigo. Acta Oto-Laryngol. 130, 1019–1023, http: //dx.doi.org/10.3109/00016481003664777. Collewijn, H., Steen, J., Ferman, L., Jansen, T.C., 1985. Human ocular counterroll: assessment of static and dynamic properties from electromagnetic scleral coil recordings. Exp. Brain Res. 59, 185–196, http://dx.doi.org/10.1007/BF00237678. Curthoys, I.S., Markham, C.H., Curthoys, E.J., 1977. Semicircular duct and ampulla dimensions in cat, guinea pig and man. J. Morphol. 151, 17–34. Dance, S., Maxey, M.R., 2003. Incorporation of lubrication effects into the forcecoupling method for particulate two-phase flow. J. Comput. Phys. 189, 212–238. Dix, M.R., Hallpike, C.S., 1952. The pathology, symptomatology and diagnosis of certain common disorders of the vestibular system. Proc. R. Soc. Med. 45, 341–354. Epley, J.M., 2001. Human experience with canalith repositioning maneuvers. Ann. N. Y. Acad. Sci. 942, 179–191, http://dx.doi.org/10.1111/j.1749-6632.2001. tb03744.x. Hall, S., Ruby, R., McClure, J., 1979. The mechanics of benign paroxysmal vertigo. J. Otolaryngol. 8. House, M.G., Honrubia, V., 2003. Theoretical models for the mechanisms of benign paroxysmal positional vertigo. Audiol. Neurootol. 8, 91–99, http://dx.doi.org/ 10.1159/000068998. Inagaki, T., Suzuki, M., Otsuka, K., Kitajima, N., Furuya, M., Ogawa, Y., Takenouchi, T., 2006. Model experiments of BPPV using isolated utricle and posterior semicircular canal. Auris Nasus Larynx 33, 129–134. Lomholt, S., Stenum, B., Maxey, M.R., 2002. Experimental verification of the force coupling method for particulate flows. Int. J. Multiph. Flow 28, 225–246. Maxey, M.R., Patel, B.K., 2001. Localized force representations for particles sedimenting in Stokes flow. Int. J. Multiph. Flow 27, 1603–1626. Obrist, D., Hegemann, S., 2008. Fluid–particle dynamics in canalithiasis. J. R. Soc. Interface 5, 1215–1229. Obrist, D., Hegemann, S., Kronenberg, D., Häuselmann, O., Rösgen, T., 2010. in vitro model of a semicircular canal: design and validation of the model and its use for the study of canalithiasis. J. Biomech. 43, 1208–1214. Oman, C.M., Marcus, E.N., Curthoys, I.S., 1987. The influence of semicircular canal morphology on endolymph flow dynamics. An anatomically descriptive mathematical model. Acta Otolaryngol. 103, 1–13. Parnes, L., Price-Jones, R., 1993. Particle repositioning maneuver for benign paroxysmal positional vertigo. Ann. Otol. Rhinol. Laryngol. 102, 325–331. Parnes, L.S., Agrawal, S.K., Atlas, J., 2003. Diagnosis and management of benign paroxysmal positional vertigo (BPPV). Can. Med. Assoc. J. 169, 681–693. Parnes, L.S., McClure, J.A., 1992. Free-floating endolymph particles: a new operative finding during posterior semicircular canal occlusion. Laryngoscope 102, 988–992, http://dx.doi.org/10.1288/00005537-199209000-00006. Rajguru, S.M., Ifediba, M.A., Rabbitt, R.D., 2004. Three-dimensional biomechanical model of benign paroxysmal positional vertigo. Ann. Biomed. Eng. 32, 831–846. Rajguru, S.M., Rabbitt, R.D., 2007. Afferent responses during experimentally induced semicircular canalithiasis. J. Neurophysiol. 97, 2355–2363, http://dx. doi.org/10.1152/jn.01152.2006. Squires, T.M., Weidman, M.S., Hain, T.C., Stone, H.A., 2004. A mathematical model for top-shelf vertigo: the role of sedimenting otoconia in BPPV. J. Biomech. 37, 1137–1146. Suzuki, M., Kadir, A., Hayashi, N., Takamoto, M., 1996. Functional model of benign paroxysmal positional vertigo using an isolated frog semicircular canal. J. Vestib. Res. 6, 121–125. Valli, P., Botta, L., Zucca, G., Valli, S., Buizza, A., 2009. Simulation of cupulolithiasis and canalolithiasis by an animal model. J. Vestib. Res.: Equilib., 89–96. Van Buskirk, W.C., Watts, R.G., Liu, Y.K., 1976. The fluid mechanics of the semicircular canals. J. Fluid Mech. 78, 87–98. Wang, H., Northrop, H., Burgess, B., Liberman, M.C., Merchant, S.N., 2006. Threedimensional virtual model of the human temporal bone: a stand-alone, downloadable teaching tool. Otol. Neurotol. 27, 452–457, http://dx.doi.org/ 10.1097/01.mao.0000188353.97795.c5.
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Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Quantitative analysis of benign paroxysmal positional vertigo fatigue under canalithiasis conditions F. Boselli a,n, L. Kleiser a, C.J. Bockisch b,c,d, S.C.A. Hegemann b, D. Obrist a a
Institute of Fluid Dynamics, ETH Zurich, 8092 Zurich, Switzerland Department of Otorhinolaryngology, Head and Neck Surgery, University Hospital Zurich, 8092 Zurich, Switzerland c Department of Neurology, University Hospital Zurich, 8092 Zurich, Switzerland d Department of Ophthalmology, University Hospital Zurich, 8092 Zurich, Switzerland b
art ic l e i nf o
a b s t r a c t
Article history: Accepted 11 March 2014
In our daily life, small flows in the semicircular canals (SCCs) of the inner ear displace a sensory structure called the cupula which mediates the transduction of head angular velocities to afferent signals. We consider a dysfunction of the SCCs known as canalithiasis. Under this condition, small debris particles disturb the flow in the SCCs and can cause benign paroxysmal positional vertigo (BPPV), arguably the most common form of vertigo in humans. The diagnosis of BPPV is mainly based on the analysis of typical eye movements (positional nystagmus) following provocative head maneuvers that are known to lead to vertigo in BPPV patients. These eye movements are triggered by the vestibulo-ocular reflex, and their velocity provides an indirect measurement of the cupula displacement. An attenuation of the vertigo and the nystagmus is often observed when the provocative maneuver is repeated. This attenuation is known as BPPV fatigue. It was not quantitatively described so far, and the mechanisms causing it remain unknown. We quantify fatigue by eye velocity measurements and propose a fluid dynamic interpretation of our results based on a computational model for the fluid–particle dynamics of a SCC with canalithiasis. Our model suggests that the particles may not go back to their initial position after a first head maneuver such that a second head maneuver leads to different particle trajectories causing smaller cupula displacements. & 2014 Elsevier Ltd. All rights reserved.
Keywords: BPPV Canalithiasis Semicircular canals (SCCs) Endolymph flow Nystagmus Method of Fundamental Solutions Force Coupling Method
1. Introduction The semicircular canals (SCCs) of the inner ear are the primary human sensors for head rotation. The SCCs are three, approximately mutually orthogonal, slender ducts which span an angle of approximately 2501 and merge into a bigger chamber called utricle (Fig. 1). One end of each SCC has a larger cross-section, the ampulla, that is plugged by a gelatinous flexible structure, the cupula. The fluid which fills the SCC, the endolymph, lags the moving canal walls during head rotation such that it deflects the cupula and the embedded hair cell bundles. This displacement and the resulting afferent signals are proportional to the head velocity, and trigger compensatory eye movements [the Vestibulo-Ocular Reflex (VOR)], which rotate the eyes opposite to the head rotation, stabilizing the eye in space and reducing visual blur. If the eye moves too far from a central position, a fast corrective movement (saccade) returns the eyes to their central position. During n
Corresponding author. E-mail addresses: [email protected] (F. Boselli), [email protected] (L. Kleiser), [email protected] (C.J. Bockisch), [email protected] (S.C.A. Hegemann), [email protected] (D. Obrist). http://dx.doi.org/10.1016/j.jbiomech.2014.03.019 0021-9290/& 2014 Elsevier Ltd. All rights reserved.
prolonged stimulation, the alternating slow compensatory and fast resetting movements are called nystagmus. A mechanical dysfunction of the SCCs, mainly of the posterior canal (PC), can lead to Benign Paroxysmal Positional Vertigo (BPPV), arguably the most common form of vertigo in humans (Baloh et al., 1989). This dysfunction is often associated with canalithiasis, a condition where free-floating particles (canaliths) reside in the SCC. Provocative head maneuvers (HM), e.g. when tilting the head backward to take a book from the top shelf, lift the particles to a higher position from which they can then settle under the action of gravity. This can lead to a flow similar to that induced by head rotations (Hall et al., 1979; Epley, 2001). The resulting post-rotatory cupula displacement triggers the misleading spinning sensation which is experienced during BPPV. The vertigo following the HM is revealed by a ‘positional’ nystagmus. This understanding of BPPV is supported by the intra-operative observation of particles in the membranous PC (Parnes and McClure, 1992), by several in-vivo models based on ampullaryafferents measurements (Suzuki et al., 1996; Inagaki et al., 2006; Rajguru and Rabbitt, 2007; Valli et al., 2009), and by several theoretical (House and Honrubia, 2003; Squires et al., 2004; Rajguru et al., 2004; Obrist and Hegemann, 2008; Boselli, 2012)
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Fig. 1. The membranous labyrinth (blue) and the bony labyrinth (yellow) of the inner ear (adapted from Obrist et al., 2010). The width of the membranous anterior (AC), posterior (PC), and horizontal (HC) canals is exaggerated for better visibility. In reality their width is only about 5% of the diameter of the bony canals (Curthoys et al., 1977). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
and in vitro (Obrist et al., 2010) models. In the case where the particles stick to the cupula rather than float in the SCC, it is common to talk about “cupulolithiasis” rather than canalithiasis. Cupulolithiasis and canalithiasis can coexist and lead to a very similar nystagmus (Cohen and Sangi-Haghpeykar, 2010). It is typical for BPPV that the positional nystagmus is generally weaker when the HM is repeated (Dix and Hallpike, 1952). This intricate phenomenon, known as BPPV fatigue, is usually attributed to the disintegration of particle lumps into smaller parts (Parnes and McClure, 1992; Parnes and Price-Jones, 1993; Parnes et al., 2003). However, previous theoretical studies do not support this mechanical interpretation of BPPV fatigue. Squires et al. (2004) concluded that a train of particles settling in a SCC would lead to a higher cupula displacement than a single particle of equal mass. Similarly, Rajguru and Rabbitt (2007) and Obrist and Hegemann (2008) repeated simulations of a HM by increasing the number of particles while keeping the total mass constant. This resulted in an increased cupula displacement rather than in ‘fatigue’. These theoretical models, however, are based on the Stokes drag force of isolated spheres and cannot account for the interaction between particles. In contrast, Boselli (2012) introduced a model based on the force coupling method (Lomholt et al., 2002) which can account for the finite size of the particles and their hydrodynamic interaction. This model was able to predict up to a 40% reduction (i.e. a fatigue) of the cupula displacement but only if the particle lump before the first HM blocks almost the whole canal lumen. For this particle size, however, the particle-driven flow is dominant already during the HM, which is in contrast to typical nystagmus measurements where the particledriven flow dominates only after the HM. Furthermore, these numerical experiments assume the resting particle position before the first and the second HM to be about the same. In the present work, we quantify the fatigue of the nystagmus in several patients and apply the model of Boselli (2012) to provide an interpretation of our nystagmus measurements based on the redistribution of particles in the SCC due to a HM.
2. Materials and methods 2.1. Nystagmus measurements The fatigue of the positional BPPV nystagmus is discussed only qualitatively in the literature. Therefore, we started by analyzing clinical nystagmus measurements of nine patients and discuss how the nystagmus changes for repeated HMs. The experimental protocol was approved by the Ethics Committee of the Canton of Zurich, Switzerland, and all patients provided written consent after the experimental procedure was explained.
Patients sat in a motorized chair that rotated the subject about an axis that intersected the head orthogonally to the plane of the posterior canal. In a first head maneuver (HM1), the rotation was backwards by 1201. The rotation was done with a constant acceleration of 101/s2 for half of the movement (601) and then decelerated with 101/s2 for the second half of the movement. This yields a triangular velocity profile with a peak of about 351/s and a total duration of about 7 s (Fig. 2). The HM1 was then reversed, and the patient brought back upright. A second head maneuver (HM2), identical to HM1, was then applied in order to assess the fatigue of the nystagmus. Before applying any rotation, we waited until the patient's nystagmus and feelings of dizziness stopped (typically at least 60 s). The orientation of the SCC before and after these rotations is illustrated in Fig. 4. For two patients, a third HM was also applied after reversing HM2. Three-dimensional eye movements were measured by the scleral search coil technique (Collewijn et al., 1985), and the rotational velocity in the plane of the affected canal was calculated. The nystagmus was measured only during and after the backward rotations. A reference light at gaze straight ahead was turned on at the start of the head rotation, and was turned off at the end. While this light suppressed eye movements during the head movement, distinguishing BPPV related nystagmus from the actual VOR would be difficult in any case. Saccades were removed with an interactive computer program that automatically detected saccades when velocity exceeded a threshold (typically about 201/s, depending upon the noise) above the median eye velocity calculated over a one second window. The automatically marked saccades could be manually adjusted and blink artifacts removed. The median velocity of each slow phase was calculated (Bockisch et al., 2013, 2012), and the resulting velocity trace was smoothed with a locally weighted second degree polynomial (smooth.m with a 10 s span, Mathworks). All patients were diagnosed with posterior canal BPPV on examination and a positive response to a Dix–Hallpike maneuver. The patients had no other neurologic or eye movement disorders. Five patients had BPPV of the left posterior canal, and four of the right posterior canal. 2.2. Numerical model The endolymph flow is modeled by the quasi-steady Stokes equations ∇pðx; tÞ þ μ∇2 uðx; tÞ ¼ f c þ f α þ f p
ð1aÞ
∇ uðx; tÞ ¼ 0
ð1bÞ
with no-slip boundary conditions at the wall; t is the time, x ¼ ðx1 ; x2 ; x3 Þ are the Cartesian coordinates of the rotating reference frame, p and u ¼ ðu1 ; u2 ; u3 Þ are the pressure and the velocity of the fluid, respectively. The inertial force f α ðx; tÞ ρα€ x arises from the angular acceleration α€ of the SCC during the head rotation (e.g. Boselli et al., 2013b; Oman et al., 1987); the force f c is the restoring force of the cupula on the fluid; and f p represents the forces exchanged between the particle and the fluid phases. Cupula model: The force f c is modeled as in the previous works (e.g. Oman et al., 1987) by introducing a pressure difference ΔP c ðtÞ ¼ KV c ðtÞ across the cupula which is proportional to the volume Vc displaced by the HM and/or the particles, where K is the stiffness of the cupula. The volumetric displacement Vc(t) of the cupula Z t ZZ u nc dA dt ð2Þ V c ðtÞ ¼ Ac
0
is computed numerically by time integration of the flow rate at a cross-section Ac of the SCC, where nc is the normal vector on the surface Ac (Boselli et al., 2013b). Fictitious forces: We adopt the force coupling method (FCM) and model each particle as a finite-size monopole (Maxey and Patel, 2001) such that f p becomes np
f p ðx; tÞ ¼ ∑ Fξ ΔM ðx Y ξ ðtÞÞ ξ¼1
ð3Þ
where Y ξ is the center of the ξ-th particle and np is the number of particles. The monopole strength Fξ is given by the sum of the volumetric forces acting on the particle and includes the gravity force, attraction/repulsive forces between the particles, and wall-particle lubrication forces (Dance and Maxey, 2003). The lubrication force, if included, is precomputed for a single particle to match asymptotic solutions (Boselli, 2012). The sliding velocity of the particle at the wall depends on the minimum distance (lubrication gap) that is imposed between the particle and the wall. The force Fξ is distributed over the particle volume by the force envelope ΔM " # −3=2 −ðx−Y ξ Þ2 ΔM x−Y ξ ¼ 2πσ 2M exp ; ð4Þ 2σ 2M pffiffiffiffi with the length scale sM ¼ ap = π similar to the particle radius ap . The fluid domain is extended over the particle volume and the velocity vξ of each particle is given by the weighted average of u at the particle location Z 3 uðx; tÞΔM ðx Y ξ ðtÞÞ d x: ð5Þ vξ ðtÞ ¼ R3
Eqs. (1) and (5) are solved by coupling the multilayer method of fundamental solutions (Boselli et al., 2012) to the FCM as proposed by Boselli et al. (2013a). We
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Fig. 2. Clinical data: Total velocity of the slow-phase positional nystagmus during ( 7 so t o 0 s) and after (0 s o t o 45 s) the first ( ), the second ( ) and the third HM ( ; only for (d) and (f)). The angular velocity (in degrees per second) of the head ( ) is shown as control. The reference light is switched off at t 1 s. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.) constrain the particle in the slender part of the posterior canal (PC), where we assume the SCC lumen to have a constant radius ascc ¼ 182 μm. The Reynolds number is small such that the local curvature of the canal can be neglected. This allowed us to map the PC geometry to a torus such that we can employ the computationally efficient domain decomposition approach proposed by Boselli et al. (2013a) for a toroidal domain. The mapping is performed by defining the position of each particle and the direction of the forces with respect to the centerline and its tangent vector, respectively. We consider the PC geometry shown in Fig. 4. Its centerline was obtained by interpolation of a projection of the anatomical model by Wang et al. (2006) in the plane of the PC.
3. Results 3.1. Nystagmus measurements Fig. 2 shows eye velocity traces for 6 of our 9 patients. Time is measured with respect to the end of the head maneuver (in green). Eye velocity typically rapidly increased after the head rotation, usually reaching a peak within 10 s of the end of the movement.
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Even-though a fixation light was present during the head movement, eye velocity begins to build up during the head movement in some
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Fig. 3. Model results: Volumetric cupula displacement Vc for the repeated head maneuver illustrated in Fig. 4; np ¼ 1 and ap ¼ 28:4 μm. Dotted lines indicate the end of each head-maneuver. Arrows indicate the peak value (Vmax) of the postrotatory cupula displacement following each head rotation. The green line shows the control (no particles). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
patients. Note, however, that this velocity could be the normal VOR response to the head movement, and not a reaction to particle movement in the canals. Only the eye velocity after the head movement can be definitively attributed to BPPV. Fatigue was clearly observed for six out of nine patients. The observed percentage reduction of the peak velocity of the positional nystagmus (t 40 s) after the second HM varies from 10 to 70% (e.g. Fig. 2(a)–(c)). For one patient, no nystagmus was observed after the second HM (100% reduction; not shown). Interestingly, for two patients (Fig. 2(e) and (f)), the positional nystagmus after the first HM shows multiple peaks. In some cases, fatigue is not clearly observed, but the peak velocity may slightly increase (Fig. 2(d)–(f)). In one of these patients (Fig. 2(d)), the second HM led to a 10% reduction of the peak velocity of the positional nystagmus, while the third HM led to an increase of about 10% (with respect to HM1). For most of the cases, we do not observe any onset latency of the positional nystagmus. This suggests that the particle-driven flow becomes significant already during the slow HM. 3.2. Numerical simulations 3.2.1. Change in the initial position due to SCC geometry The same HMs imposed during the measurements (Fig. 2, green) were implemented in our computational model. The predicted cupula
Fig. 4. Orientation of the PC (green) before (left; patient upright) and after (right) the first (top) and the second (bottom) head maneuver (HM). The red sphere illustrates the position of the particle at the beginning (left) and at the end (right) of the HM. A, B, and C indicate the initial position of the particle before each head rotation (i.e. HM1, reversed HM1, and HM2). Circular arrows indicate the direction of head rotation. Lines illustrate the particle trajectory with arrows indicating the particle motion. The bony labyrinth (topleft; gray) is shown for illustrative purposes only. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
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displacement Vc computed for one particle of radius ap ¼ 28:4 μm is shown in Fig. 3 together with a control (result for a healthy SCC without particles). Note that the considered HM is very slow such that the SCC operates close to its lower corner frequency (Van Buskirk et al., 1976). Therefore, the cupula displacement for the control does not exactly follow the triangular shape of the imposed angular velocity, but we observe a large overshoot toward the end of the HM. When the first HM (HM1) starts, it is reasonable to assume that the particle is close to the ampulla (Fig. 4, position A), at the lowest point of the PC, when the subject is upright. During HM1, the particle is lifted to a higher position from which it settles toward the lower part of the canal, away from the ampulla. In agreement with the common understanding of canalithiasis (Hall et al., 1979; Epley, 2001), this leads to a temporary, particle-driven flow which results in a misleading, ampullofugal (V c o 0) post-rotatory cupula displacement while in fact the head is at rest (Fig. 3). This post-rotatory displacement arises after an onset-latency which is required to compensate for the cupula overshoot observed at the end of the HM for the control (Obrist and Hegemann, 2008). Reversing the HM1 to bring the patient back upright, the particles settle back, leading to a weaker, ampullopetal (V c 4 0), i.e. inhibiting, post-rotatory cupula displacement (Fig. 3; not measured in vivo). The particle may settle at a position (Fig. 4, position B) which is farther from the ampulla than its initial position A. This shift in the initial particle position is favored by the fact that the canal centerline between points A and
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B is almost straight. Starting from position B rather than A, the particle is lifted to a lower position during HM2. Settling from this lower position (Fig. 4), the particle leads to a weaker post-rotatory cupula displacement with a peak value Vmax which is about 70% smaller than what has been observed after HM1 (cf. Fig. 3). This reduction can explain the fatigue of the positional nystagmus observed in our patients. Simulations were repeated for several values of particle mass, size ap and number np, and with and without lubrication force (Table 1). The reduction of Vmax due to repeated HMs ranges between 25% and 84% for the presented simulations, and decreases by increasing either ap, np, or particle mass (Table 1). Increasing the particle-wall friction by adding the lubrication force tends to lift the particles to a higher position and to enhance Vmax. Fatigue is also enhanced (Table 1) because it increases the difference in the height to which the particles are lifted by HM1 and HM2, respectively. To test the effect of small changes in the head position, some simulations were repeated with the SCC rotated by 101 (i.e. position A lower and point B higher at t¼ 0). The particles reach a lower position after HM1 and lead to a slightly lower value of Vmax, while cupula displacement following HM2 is unchanged. This tends to slightly weaken fatigue. In our simulations with multiple particles, the particles line up near position B when HM2 starts (Fig. 5(a), top). However, the actual position of the particles when HM2 starts will depend on the precise SCC orientation and particle–wall interaction.
Table 1 Percentage reduction of the maximum post-rotatory cupula displacement due to a change in the particles initial position from about A (HM1) to about B (HM2). The particles sizes are chosen such that the total particle mass is constant along the diagonals of this table.
np ¼ 1 np ¼ 2 np ¼ 4 np ¼ 8
ap ¼ 14:2 μm
ap ¼ 17:89 μm
– – – 30%
– – 80% –
(a) Results with lubrication force ap ¼ 22:54 μm ap ¼ 28:4 μm – 84% – –
74.1% 73.22% 67.34% –
ap ¼ 35:78 μm
ap ¼ 45:08 μm
71.6% – – –
25% – – –
(b) Results without lubrication force
np ¼ 1 np ¼ 2 np ¼ 4 np ¼ 8 np ¼ 16
ap ¼ 11:27 μm
ap ¼ 14:2 μm
ap ¼ 17:89 μm
ap ¼ 22:54 μm
ap ¼ 28:4 μm
ap ¼ 35:78 μm
– – – – 75%
– – – 71% 67%
– – 69% 65% –
– 57% 38% – –
37% 34% – – –
32% – – – –
0.5
x 10
−12
0
−1
c
V [m3]
−0.5
−1.5 −2 −2.5 −3 −10
−5
0
5
10
15
20
25
t [s] Fig. 5. (a) Different initial positions of the particles for the simulations with np ¼10 and ap ¼ 0:13ascc . (b) Volumetric cupula displacement for the different initial positions defined in (a): , close to the ampulla (position A); , away from ampulla (position B); y, dispersed between position A and B.
F. Boselli et al. / Journal of Biomechanics 47 (2014) 1853–1860
Therefore, we tested another possible scenario where the particles are dispersed along the horizontal segment AB of the canal (Fig. 5 (a)) before HM2 starts. For such a scenario, we present results for np ¼10 and ap ¼ 0:13ascc (Fig. 5(b)). A first HM is simulated by assuming that the particles are initially close to the ampulla about position A (Fig. 5(a), top). The particle-driven flow becomes dominant during the HM, such that no onset-latency is observed. The volumetric cupula displacement shows two welldistinguishable peaks. These are associated with the formation of two groups of particles that detach from the inner wall at two different times (cf. Fig. 4, top right). If we assume the particles to be dispersed between the positions A and B (Fig. 5(a), center) when HM2 starts, the presence of two separate peaks becomes less pronounced in the resulting cupula displacement. This result is similar to what we observe in some of our nystagmus measurements (e.g. Fig. 2(e)), where the double peak was observed only after HM1. Particles positioned about position B (Fig. 5(a)) before the HM lead to a weaker nystagmus because of the larger average initial distance of the particles from the ampulla. 3.2.2. Particle lump disintegration In order to replicate the disintegration of the particle lump suggested by Parnes and McClure (1992), we present an example (Fig. 6(a)) where we add an attraction force between the particles in the form of a Lennard-Jones potential (e.g. Castellanos, 2005). When the particles are at rest, they lump together because of this attraction force. During the first HM, the lump breaks and the particles disperse while settling in the SCC (Fig. 6(b)). The resulting post-rotatory cupula displacement shows a double-peak cupula displacement that resembles the nystagmus velocity in Fig. 2(e). Before HM2 starts, the particles tend to align along the wall in a single file and come to rest around position B. From this position, the particles lead to a weaker cupula displacement after HM2, as for the simulations without attraction force. The lubrication gap for this example was set to 10 6 ap , and the lubrication force was precomputed for ap ¼ 0:13ascc . In order to estimate the effect of particle disintegration alone (with the particles moving back to position A after HM1), we can compare the cupula displacement following HM1 obtained with and without attraction force (Fig. 6(a)). In this example, the attraction force reduces the predicted Vmax. This supports the idea that disintegration of a particle lump alone is not the only explanation for fatigue. Rather the particles have to move further away from the cupula. 3.2.3. Exceptions not leading to fatigue In order to identify conditions that could explain cases where fatigue is not observed, we considered the SCC after a backward rotation by 1201 (e.g. after HM1) and plot the cupula displacement induced by a particle settling from different elevated positions. Examples are shown in Fig. 7. Based on the initial height of the particle, we can identify three types of particle trajectories: type I, II and III. A particle following the type I trajectory settles first along the inner side of the wall before detaching again from the wall and reattaching at the outer side (Fig. 7, trajectories 1–4). A particle following a type II trajectory detaches significantly from the outer wall, but does not slide on the inner wall (Fig. 7, trajectories 6 and 7). A particle following a type III trajectory detaches only slightly from the outer wall and reattaches again at the outer wall, or simply settles along the outer wall without any detachment from the wall (Fig. 7, trajectory 8). Fig. 7(b and c) shows that Vmax does not decrease monotonically with decreasing initial altitude of the particle. Instead, Vmax can increase when the initial altitude is decreased in the narrow range where we observe the transition between type I and
−13
x 10
2 0 Vc [m3]
1858
−2 −4 −6 −8 −10
0
10
20
30 t [s]
40
50
60
Fig. 6. (a) Volumetric cupula displacement during and after the first (black) and the second (red) HM for np ¼8 and ap ¼ 15 μm. Solid lines: Simulation with attraction force; dashed line: Simulation without attraction force. (b) Snapshots of the particles during and after HM1 for the simulation with the attraction force. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
type II trajectories (Fig. 7, trajectories 4–6). This suggests that the post-rotatory cupula displacement after HM2 can increase if the particles follow a type I trajectory after HM1 and a type II trajectory after HM2. This provides a possible explanation for the more rare cases where a slight increase of the eye velocity is observed when repeating the HM (cf. Fig. 2(d) and (f)). Furthermore, the time to peak tp (Fig. 7(c)) tends to decrease with decreasing initial altitude for type I trajectories (Fig. 7, trajectories 1–5), while it increases for type II and III trajectories (Fig. 7, trajectories 6–8).
4. Discussion and conclusions We presented nystagmus measurements for repeated head maneuvers (HMs) which provide a quantitative reference for BPPV fatigue. The reduction in cupula displacement leading to
F. Boselli et al. / Journal of Biomechanics 47 (2014) 1853–1860
x 10
−12
3
|Vmax| [m ]
Type I
Type II
Type III
1.6 1.4 1.2 1
1
2
3
3.5
4 5 trajectory number
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6
Type II
7
8
Type III
3
p
t [s]
1859
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1
2
3
4 5 trajectory number
6
7
8
Fig. 7. The PC is held fixed in its position at the end of the head maneuver and a particle is released at t¼ 0 from different initial positions which leads to different trajectories (numbered from 1 to 8). The cupula displacement Vc at t ¼0 is set to zero. (a) Particle trajectories; (b) volumetric cupula displacements Vc; (c) peak volumetric cupula displacement Vmax and time to peak tp. No lubrication force is considered; ap ¼ 36:8 μm.
the measured decrease in eye velocity was predicted by means of a numerical model of particle–fluid dynamics in the SCC. The presented results suggest that the fatiguability of the nystagmus can be explained by the fact that a first HM tends to move the particles farther from the ampulla. This is favored by the PC anatomy which may feature a ‘straight’ part close to the ampulla. We also modeled disintegration of a particle lump by introducing an attraction force between particles. This condition was suggested to represent the actual scenario in vivo (Parnes and McClure, 1992). However, fatigue is enhanced when the attraction force is not considered, suggesting that disintegration of particle lumps is not necessary for fatigue and that fatigue is dominated by changes in the position of the particles following a first HM. Fatigue was not observed for 3 out of 10 patients, for which the positional nystagmus was even increased when repeating the HM.
For these patients, the particles might have gone back to their initial position after HM1 (e.g. due to a particular SCC shape). A second possibility is that the particle initial position for the two consecutive HMs is such that the particle trajectory changes from type I to type II trajectory as discussed in Section 3.2.3 (e.g. from trajectory 4 to 6 in Fig. 7), which would lead to an increase of the cupula displacement following HM2. A further possibility (not modeled) is that these patients were initially affected by cupulolithiasis. This would prevent fatigue because the particles cannot move away from the cupula. In such a scenario, if one of the particles detaches from the cupula following HM1, it will generate a canalithiasis on top of the cupulolithiasis (Cohen and SangiHaghpeykar, 2010). This could increase the nystagmus following HM2, because the same particle leads to a bigger transcupular pressure under canalithiasis rather than cupulolithiasis (Squires et al., 2004; Rajguru et al., 2004).
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Besides the actual fatigue, our computational model could reproduce the multiple peaks of the positional nystagmus observed in two of our patients. This feature is associated with the different instants at which the different particles detach from the wall. In some patients the time to peak tp seems to decrease after the first HM. This feature was not predicted by our model. We associate this discrepancy with the fact that our model assumes a uniform radius of the SCC, while the cross sectional area increases toward the cupula of the actual SCC. Squires et al. (2004) showed that a particle does not affect the cupula significantly while settling in a larger region. We can then imagine that the particles in position A settle first along a larger region during HM1 such that their action on the cupula is delayed. In contrast, the particles are already in the narrow part of the SCC (position B) when HM2 starts, and they immediately become effective in displacing the cupula when they start settling such that tp is shorter after HM2 than after HM1. A natural next step for our model would be the employment of patient-specific geometries, including the non-planarity of the SCC. Recent advances in the field of vestibular anatomy (Bradshaw et al., 2010) suggest that this might become possible in the near future. Conflict of interest statement All authors declare that there are no conflicts of interest. Acknowledgment The work of F.B. was supported by the Swiss National Science Foundation (SNF # 200021-116575). References Baloh, R.W., Sloane, P.D., Honrubia, V., 1989. Quantitative vestibular function testing in elderly patients with dizziness. Ear Nose Throat J. 68, 935–939. Bockisch, C.J., Khojasteh, E., Straumann, D., Hegemann, S.C.A., 2012. Development of eye position dependency of slow phase velocity during caloric stimulation. PLoS ONE 7, e51409, http://dx.doi.org/10.1371/journal.pone.0051409. Bockisch, C.J., Khojasteh, E., Straumann, D., Hegemann, S.C.A., 2013. Eye position dependency of nystagmus during constant vestibular stimulation. Exp. Brain Res. 226, 175–182, http://dx.doi.org/10.1007/s00221-013-3423-6. Boselli, F., 2012. Fluid Dynamics of the Balance Sense: A Computational Study Based on the Multilayer Method of Fundamental Solutions (Ph.D. thesis). ETH Zurich, Zurich, Switzerland. Diss. ETH No. 20576. Boselli, F., Obrist, D., Kleiser, L., 2012. A multilayer method of fundamental solutions for Stokes flow problems. J. Comput. Phys. 231, 6139–6158, http://dx.doi.org/ 10.1016/j.jcp.2012.05.023. Boselli, F., Obrist, D., Kleiser, L., 2013a. A meshless boundary method for Stokes flows with particles: application to canalithiasis. Int. J. Numer. Methods Biomed. Eng. 29, 1176–1191, http://dx.doi.org/10.1002/cnm.2564. Boselli, F., Obrist, D., Kleiser, L., 2013b. Vortical flow in the utricle and the ampulla: a computational study on the fluid dynamics of the vestibular system. Biomech. Model. Mechanobiol. 12, 335–348, http://dx.doi.org/10.1007/s10237-012-0402-y. Bradshaw, A., Curthoys, I., Todd, M., Magnussen, J., Taubman, D., Aw, S., Halmagyi, G., 2010. A mathematical model of human semicircular canal geometry: a new
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